# York University Applied and Industrial Mathematics Seminar

## LIAM Summer Program 2005

Organizers: Jianhong Wu and Huaiping Zhu
The program is set to take advantage of the presence of many distinguished visitors to LIAM during the summer of 2005, and to provide a framework for our students to interact with each other. It consists of a seminar series, some one-day events such as symposia, and a graduate-student day.

# Mini-symposium on Nonlinear Dynamics

### Organizers: Jianhong Wu and Huaiping Zhu

10:30-11:30: Prof. Hildebrando Munhoz Rodrigues
Institution: Instituto de Ciencias Matematicas e de Computacao,
University of Sao Paulo, Brasil

TITLE: Smooth Linearization in Infinite Dimensional Dynamical Systems.
ABSTRACT: I will present some recent results on this subject, obtained in joint works with Prof. J. Sola-Morales from Universitat Politecnica de Catalunya, Spain. Under certain assumptions, including a nonresonance condition, we proved that smooth linearization is possible in the case of contractions and in a case of saddle. These results extend to infinite dimensions some classical results proved by Philip Hartman in finite dimensions. I will also present an example of an invertible analytic contraction defined in a Hilbert space that is not C1-linearizable. In this example the nonresonance conditions are not satisfied.
E-mail: hmr@icmc.usp.br

11:30-12:30: Prof. Xiang Zhang
Department of Mathematics,
Shanghai Jiaotong University, China

TITLE: Embedding diffeomorphism in differential flows
ABSTRACT: In this talk, we introduce my joint work with Weigu Li and Jaume Llibre on the embedding diffeomorphism in differential flows. In higher dimension spaces, J.Palis pointed out that the diffeomorphism admitting an embedding vector field is few in the Baire sense. We provide a sufficient condition in order that a diffeomorphism can be embedded in a differential vector field. The idea of the proof is related to the extension of Floquet's theory, and the relation between equivalence of vector fields and the conjugacy of the corresponding Poincare map.
Webpage: Prof. Xiang Zhang
E-mail: xzhang@sjtu.edu.cn

12:30-2:30: Lunch Break

2:30-3:30: Prof. Victor Vlasov
Department of Mathematics,
Moscow State University

TITLE: Spectral Problems arising in the theorey of differential equations with delay
ABSTRACT: We present a review of results on asymptotic behavior and stability of strong solutions for functional differential equations ( FDE ). We also formulate the results about spectral properties ( completeness and basisness ) of exponenntial solutions of above mentioned equations. It is important to emphasize that our approach for the researching of FDE is based on the srectral operator analysis .
E-mail: vicvvlasov@rambler.ru

3:30-4:30: Prof. Linghai Zhang
Department of Mathematics,
Lehigh University, USA

TITLE: Traveling waves of a singularly perturbed system of integral-differential equations arising from neuronal networks
ABSTRACT: A nonlinear nonlocal model arising from synaptically coupled neuronal networks with two integral terms is considered. The existence and stability of several traveling wave solutions are established by using ideas in differential equations and functional analysis. Steady-state solutions of inhomogeneous integral-differential equations are also investigated. We consider several types of kernel functions: (I) positive functions, such as $K(x)=\frac\rho2\exp(-\rho|x|)$ and $K(x)=\sqrt{\frac\rho\pi}\exp(-\rho x^2)$, where $\rho>0$ is a constant; (II) nonnegative kernels with compact supports, for examples, (i) $K(x)=\frac12$ for $|x|\x1$ and $K(x)=0$ for $|x|>1$, and (ii) $K(x)=\frac12\cos x$ for $|x|\x\frac\pi2$ and $K(x)=0$ for $|x|>\frac\pi2$; (III) Mexican hat type kernel functions such as $K(x)=A\exp(-a|x|)-B\exp(-b|x|)$ and $K(x)=A\exp(-ax^2)-B\exp(-bx^2)$, where $A>B>0$ and $a>b>0$ are constants.
E-mail: liz5@lehigh.edu

Seminar 7: August 3, Wednesday, 11:30-12:30, N638 Ross

SPEAKER: Prof. Yanping Chen
Department of Mathematics, Xiantan University

TITLE: The mixed finite element method for convex optimal control problems

ABSTRACT:

In this talk, we investigate the full discretization of general convex optimal control problems using mixed finite element methods. We derive error estimates, sharp posteriori error estimates and superconvergence results for both the control and state approximations. Moreover, we study the numerical approximation of convex optimal control problems with oscillating coefficients by using a mixed multiscale finite element method.

E-mail: ypchen@xtu.edu.cn

Seminar on Chaos: July 29, Friday, 2:30-4:30, N638 Ross

2:30-3:30 SPEAKER: Prof. Guanrong Chen
Department of Electronic Engineering, City University of Hongkong

TITLE: Beyond the Lorenz System

ABSTRACT:

In this talk, the Generalized Lorenz Canonical Form of a large family of chaotic systems will be introduced. This canonical form includes the classical Lorenz system and the recently dis-covered Chen system as two extreme cases in a train of infinitely many related but not equiva-lent chaotic systems. The main idea in developing this new unified theory will be explained, computer simulation and circuit implementation will be demonstrated, and mathematical analysis will be briefly outlined. Moreover, information about some real applications of the Chen system in image encryption and liquid mixing will be given. Finally, the recently developed hyperchaotic Chen system will be demonstrated and discussed.

Webpage:: Prof. Guanrong Chen

E-mail: eegchen@cityu.edu.hk
Prof. Chen received the MSc degree in Computational Mathematics from Zhongshan University, China in 1981 and the PhD degree in Applied Mathematics from Texas A&M University in 1987. He currently is a Chair Professor in the Department of Electronic Engineering and the Founding Director of the Centre for Chaos Control and Synchronization at the City University of Hong Kong. He is a Fellow of the IEEE (1996). His research interest is within the fields of nonlinear dynamics and con-trols, complex networks, and applications of nonlinear science in system engineering and information technology.

3:30-4:30 SPEAKER: Prof. Yuming Shi
Department of Mathematics, Shandong University, China

TITLE: Some Criteria of Discrete Chaos and Their Applications

ABSTRACT:

In this talk, some recently obtained results on discrete chaos will be presented. First, three definitions of chaos, namely, the Li-Yorke chaos, Wiggins chaos, and Devaney chaos, will be introduced. Then, their relationships will be discussed, followed by some rigorous criteria of chaos for general discrete dynamical systems in finite-dimensional spaces, Banach spaces, and general complete metric spaces. Finally, some applications of the new results in anticontrol of chaos will be discussed.

Prof. Yuming Shi
Prof. Shi received the MSc degree in Mathematics from Nankai University, China in 1990 and the PhD degree in Mathematics from Shandong University, China in 2000. She is now a full professor in the Department of Mathematics at Shandong University. Her research interest is within the fields of chaos theory and its applications, and spectral theory of differential and difference operators.

Seminar 4: July 28, Thursday, 11:30-12:30, N638 Ross

SPEAKER: Prof. Elena Braverman
Department of Mathematics, University of Calgary

TITLE: On stability of delay differential equations with variable delays and coefficients

ABSTRACT:

In the first part of the talk, some new results on stability of linear delay equations with several delays and variable delays and coefficients are presented. They are based on the comparison to some model equation with well known stability properties. These results can also be applied to the local stability of nonlinear equations.

As an example, we consider the Mackey-Glass equation with variable coefficients and a nonconstant delay N'=[r(t)N(g(t))]/[1+(N(g(t))^c] - b(t)N(t) which models white blood cells production. Qualitative properties of this equation, such as boundedness of solutions, persistence and oscillation, are discussed. It is also demonstrated that with the delay in the mortality term as well, the solution may become negative.

Webpage:: Prof. Elena Braverman
E-mail: maelena at math.ucalgary.ca

Seminar 3: July 27, Wednesday, 4:00-5:00, N638 Ross

SPEAKER: Prof. Kazufumi Ito
North Carolina State University

TITLE: Applications of Semi-smooth Newton method to Variational Inequalities

ABSTRACT:

In This talk semismooth Newton methods for solving nonlinear non-smooth equations $F(x)=0$ in Banach spaces are discussed. These investigations are motivated by complementarity problems, variational inequalities and optimal control problems with control or state constraints, for example. Assuming semismoothness of it is shown that the superlinearly convergent Newton method can be globalized, if the merit function $|F(x)|^2$ has appropriate descent directions..

Webpage:: Prof. Kazufumi Ito
E-mail: kito@math.ncsu.edu

Seminar 2: July 13, 2:00-3:00, Ross N638

SPEAKER: A. Zaratsyan
Universite de Montreal.

TITLE: Discrete and continuous cosine transform generalized to compact semisimple Lie groups of rank two

ABSTRACT:

Co-author: J. Patera, Universite de Montreal
We develop and describe continuous and discrete transforms of functions on compact semisimple Lie groups as their expansions into series of uncommon special functions, called here "C-functions" (in recognition of the fact that the functions generalize the cosine to any finite dimension). Discretization on lattices of any density is described and the continuous and discrete orthogonality of C-functions is shown. It is shown that one-dimensional discrete transform coincides with the cosine transform, and, in addition, four variants of the transform in the two-dimensional case are presented.

Seminar 1: June 8, Wednesday, 3:00-4:00, Ross N638

SPEAKER: Prof. Liping Liu
Department of Mathematics, University of Texas Pan American

TITLE: Nonlinear Structure Models

ABSTRACT:

The interaction of a flexible structure with a flowing liquid in which it is submersed or by which it is surrounded gives rise to a rich variety of physical phenomena with applications in many fields of engineering, for example, the stability and response of aircraft wings, the flow of blood through arteries, the response of bridges and tall buildings to winds, the vibrations of turbine and compressor blades, and the oscillation of heat exchange. To understand these phenomena we need to model both the structure and the fluid. Furthermore, the nonlinear effects may come from structure and/or fluid. In this talk, the primary emphasize is on the nonlinear structure models. The applications are largely drawn from aerospace engineering, although the methods and the fundamental phenomena have much wider applications. Some video clips of the experiments mounted in the low speed wind tunnel at Duke University will be shown, followed by the corresponding equation models. Some of our recent results will also be shown and explained therein.

Webpage::
E-mail: